Cyclically five-connected cubic graphs

A cubic graph $G$ is cyclically 5-connected if $G$ is simple, 3-connected, has at least 10 vertices and for every set $F$ of edges of size at most four, at most one component of $G\backslash F$ contains circuits. We prove that if $G$ and $H$ are cyclically 5-connected cubic graphs and $H$ topologically contains $G$, then either $G$ and $H$ are isomorphic, or (modulo well-described exceptions) there exists a cyclically 5-connected cubic graph $G'$ such that $H$ topologically contains $G'$ and $G'$ is obtained from $G$ in one of the following two ways. Either $G'$ is obtained from $G$ by subdividing two distinct edges of $G$ and joining the two new vertices by an edge, or $G'$ is obtained from $G$ by subdividing each edge of a circuit of length five and joining the new vertices by a matching to a new circuit of length five disjoint from $G$ in such a way that the cyclic orders of the two circuits agree. We prove a companion result, where by slightly increasing the connectivity of $H$ we are able to eliminate the second construction. We also prove versions of both of these results when $G$ is almost cyclically 5-connected in the sense that it satisfies the definition except for 4-edge cuts such that one side is a circuit of length four. In this case $G'$ is required to be almost cyclically 5-connected and to have fewer circuits of length four than $G$. In particular, if $G$ has at most one circuit of length four, then $G'$ is required to be cyclically 5-connected. However, in this more general setting the operations describing the possible graphs $G'$ are more complicated.Comment: 47 pages, 5 figures. Revised according to referee's comments. To appear in J. Combin. Theory Ser.

K-6 minors in large 6-connected graphs

Jorgensen conjectured that every 6-connected graph with no K-6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs. (C) 2017 Published by Elsevier Inc

Declarative Specification

Deriving formal specifications from informal requirements is extremely difficult since one has to overcome the conceptual gap between an application domain and the domain of formal specification methods. To reduce this gap we introduce application-specific specification languages, i.e., graphical and textual notations that can be unambiguously mapped to formal specifications in a logic language. We describe a number of realised approaches based on this idea, and evaluate them with respect to their domain specificity vs. generalit

Boundary operator algebras for free uniform tree lattices

Let $X$ be a finite connected graph, each of whose vertices has degree at least three. The fundamental group $\Gamma$ of $X$ is a free group and acts on the universal covering tree $\Delta$ and on its boundary $\partial \Delta$, endowed with a natural topology and Borel measure. The crossed product $C^*$-algebra $C(\partial \Delta) \rtimes \Gamma$ depends only on the rank of $\Gamma$ and is a Cuntz-Krieger algebra whose structure is explicitly determined. The crossed product von Neumann algebra does not possess this rigidity. If $X$ is homogeneous of degree $q+1$ then the von Neumann algebra $L^\infty(\partial \Delta)\rtimes \Gamma$ is the hyperfinite factor of type $III_\lambda$ where $\lambda=1/{q^2}$ if $X$ is bipartite, and $\lambda=1/{q}$ otherwise

New Lower Bounds for Some Multicolored Ramsey Numbers

We use finite fields and extend a result of Fan Chung to give eight new, nontrivial, lower bounds.Comment: 6 page